Influence of an $L^p$-perturbation on Hardy-Sobolev inequality with singularity a curve

Abstract

We consider a bounded domain $\Omega$ of $\mathbb{R}^N$, $N\ge3$, $h$ and $b$ continuous functions on $\Omega$. Let $\Gamma$ be a closed curve contained in $\Omega$. We study existence of positive solutions $u \in H^1_0\left(\Omega\right)$ to the perturbed Hardy-Sobolev equation: $$ -\Delta u+h u+bu^{1+\delta}=\rho^{-\sigma}_\Gamma u^{2^*_\sigma-1} \qquad \textrm{ in } \Omega, $$ where $2^*_\sigma:=\frac{2(N-\sigma)}{N-2}$ is the critical Hardy-Sobolev exponent, $\sigma\in [0,2)$, $0< \delta<\frac{4}{N-2}$ and $\rho_\Gamma$ is the distance function to $\Gamma$. We show that the existence of minimizers does not depend on the local geometry of $\Gamma$ nor on the potential $h$. For $N=3$, the existence of ground-state solution may depends on the trace of the regular part of the Green function of $-\Delta+h$ and or on $b$. This is due to the perturbative term of order ${1+\delta}$.